Our general question has now become whether \(\mathsf {Forget}: \mathsf {LLD_{sp}}\rightarrow \mathsf {MSN_{sp}}\)  admits a right adjoint.
Freyd's Adjoint Functor Theorem [1]} could prove
useful, however do not know whether it applies, mostly because we do
not know whether coequalizers exist in \(\mathsf {MSN_{sp}}\) .
We gather in Table REF  the information that we know about
limits and colimits in the three categories we introduced.
<TABLE>